Abstract

We show how to adapt an efficient numerical algorithm to
obtain an approximate solution of a system of pantograph equations. This
algorithm is based on a combination of Laplace transform and Adomian
decomposition method. Numerical examples reveal that the method is
quite accurate and efficient, it approximates the solution to a very high
degree of accuracy after a few iterates.

1. Introduction

The pantograph equation:𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡),𝑢(𝑞𝑡)),𝑡≥0,𝑢(0)=𝑢0,(1.1)
where 0<𝑞<1 is one of the most important kinds of delay differential equation that arise in many scientific models such as population studies, number theory, dynamical systems, and electrodynamics, among other. In particular, it was used by Ockendon and Tayler [1] to study how the electric current is collected by the pantograph of an electric locomotive, from where it gets its name.

The primary aim of this paper is to develop the Laplace decomposition for a system of multipantograph equations:𝑢1(𝑡)=𝛽1𝑢1(𝑡)+𝑓1𝑡,𝑢𝑖(𝑡),𝑢𝑖𝑞𝑗𝑡,𝑢2(𝑡)=𝛽2𝑢2(𝑡)+𝑓2𝑡,𝑢𝑖(𝑡),𝑢𝑖𝑞𝑗𝑡,⋮𝑢𝑛(𝑡)=𝛽𝑛𝑢𝑛(𝑡)+𝑓𝑛𝑡,𝑢𝑖(𝑡),𝑢𝑖𝑞𝑗𝑡,𝑢𝑖(0)=𝑢𝑖0,𝑖=1,…,𝑛,𝑗=1,2,…,(1.2)
where 𝛽𝑖,𝑢𝑖0∈𝒞, and 𝑓𝑖 are analytical functions, and 0<𝑞𝑗<1.

In 2001, the Laplace decomposition algorithm (LDA) was proposed by khuri in [2], who applied the scheme to a class of nonlinear differential equations. In this method, the solution is given as an infinite series usually converging very rapidly to the exact solution of the problem.

A major advantage of this method is that it is free from round-off errors and without any discretization or restrictive assumptions. Therefore, results obtained by LDA are more accurate and efficient. LDA has been shown to easily and accurately to approximate a solutions of a large class of linear and nonlinear ODEs and PDEs [2–4]. Ongun [5], for example, employed LDA to give an approximate solution of nonlinear ordinary differential equation systems which arise in a model for HIV infection of CD4+ T cells, Wazwaz [6] also used this method for handling nonlinear Volterra integro-differential equations, Khan and Faraz [7] modified LDA to obtain series solutions of the boundary layer equation, and Yusufoglu [8] adapted LDA to solve Duffing equation.

The numerical technique of LDA basically illustrates how Laplace transforms are used to approximate the solution of the nonlinear differential equations by manipulating the decomposition method that was first introduced by Adomian [9, 10].

2. Adaptation of Laplace Decomposition Algorithm

We illustrate the basic idea of the Laplace decomposition algorithm by considering the following system:𝐿𝑡𝑢1=𝑅1𝑢1,…,𝑢𝑛+𝑁1𝑢1,…,𝑢𝑛+𝑔1,𝐿𝑡𝑢2=𝑅2𝑢1,…,𝑢𝑛+𝑁2𝑢1,…,𝑢𝑛+𝑔2,⋮𝐿𝑡𝑢𝑛=𝑅n𝑢1,…,𝑢𝑛+𝑁𝑛𝑢1,…,𝑢𝑛+𝑔𝑛.(2.1)

With the initial condition𝑢𝑖(0)=𝑢𝑖0,𝑖=1,…,𝑛,(2.2)
where 𝐿𝑡 is first-order differential operator, 𝑅𝑖 and 𝑁𝑖, 𝑖=1,…,𝑛, are linear and nonlinear operators, respectively, and 𝑔𝑖, 𝑖=1,…,𝑛, are analytical functions.

The technique consists first of applying Laplace transform (denoted throughout this paper by ℒ) to the system of equations in (2.1) to getℒ𝐿𝑡𝑢1𝑅=ℒ1𝑢1,…,𝑢𝑛𝑁+ℒ1𝑢1,…,𝑢𝑛𝑔+ℒ1,ℒ𝐿𝑡𝑢2𝑅=ℒ2𝑢1,…,𝑢𝑛𝑁+ℒ2𝑢1,…,𝑢𝑛𝑔+ℒ2,⋮ℒ𝐿𝑡𝑢𝑛𝑅=ℒ𝑛𝑢1,…,𝑢𝑛𝑁+ℒ𝑛𝑢1,…,𝑢𝑛𝑔+ℒ𝑛.(2.3)

Using the properties of Laplace transform, and the initial conditions in (2.2) to getℒ𝑢1=ℋ11(𝑠)+𝑠ℒ𝑅1𝑢1,…,𝑢𝑛+1𝑠ℒ𝑁1𝑢1,…,𝑢𝑛,ℒ𝑢2=ℋ21(𝑠)+𝑠ℒ𝑅2𝑢1,…,𝑢𝑛+1𝑠ℒ𝑁2𝑢1,…,𝑢𝑛,⋮ℒ𝑢𝑛=ℋ𝑛1(𝑠)+𝑠ℒ𝑅𝑛𝑢1,…,𝑢𝑛+1𝑠ℒ𝑁𝑛𝑢1,…,𝑢𝑛,(2.4)
whereℋ𝑖1(𝑠)=𝑠𝑢𝑖𝑔(0)+ℒ𝑖,𝑖=1,…,𝑛.(2.5)

The Laplace decomposition algorithm admits a solution of 𝑢𝑖(𝑡) [2] in the form𝑢𝑖(𝑡)=∞𝑗=0𝑢𝑖𝑗(𝑡),𝑖=1,…,𝑛,(2.6)
where the terms 𝑢𝑖𝑗(𝑡) are to be recursively computed. The nonlinear operator 𝑁𝑖 is decomposed as follows:𝑁𝑖𝑢1,…,𝑢𝑛=∞𝑗=0𝐴𝑖𝑗.(2.7)
and 𝐴𝑖𝑗 are the so-called Adomian polynomials that can be derived for various classes of nonlinearity according to specific algorithms set by Adomian [9, 10].𝐴𝑖0𝑢=𝑓𝑖0,𝐴𝑖1=𝑢𝑖1𝑓′𝑢𝑖0,𝐴𝑖2=𝑢𝑖2𝑓𝑢𝑖0+1𝑢2!2𝑖1𝑓𝑢𝑖0,𝐴𝑖3=𝑢𝑖3𝑓𝑢𝑖0+𝑢𝑖1𝑢𝑖2𝑓𝑢𝑖0+1𝑢3!2𝑖1𝑓𝑢𝑖0,….(2.8)
Substituting (2.6) and (2.7) into (2.4),and Using the linearity of Laplace transform, we get∞𝑗=0ℒ𝑢1𝑗=ℋ1(1𝑠)+𝑠∞𝑗=0ℒ𝑅1𝑢1𝑗,…,𝑢𝑛𝑗+1𝑠∞𝑗=0ℒ𝐴1𝑗,∞𝑗=0ℒ𝑢2j=ℋ2(1𝑠)+𝑠∞𝑗=0ℒ𝑅2𝑢1𝑗,…,𝑢𝑛𝑗+1𝑠∞𝑗=0ℒ𝐴2𝑗,⋮∞𝑗=0ℒ𝑢𝑛𝑗=ℋ𝑛1(𝑠)+𝑠∞𝑗=0ℒ𝑅𝑛𝑢1𝑗,…,𝑢𝑛𝑗+1𝑠ℒ∞𝑗=0𝐴𝑛𝑗.(2.9)

We thus have the following recurrence relations from corresponding terms on both sides of (2.9):
ℒ𝑢𝑖𝑜(𝑡)=ℋ𝑖ℒ𝑢(𝑠),(2.10)𝑖1=1(𝑡)𝑠ℒ𝑅𝑢10,…,𝑢𝑛0+1𝑠ℒ𝐴𝑖0ℒ𝑢,(2.11)𝑖2=1(𝑡)𝑠ℒ𝑅𝑢11,…,𝑢𝑛1+1𝑠ℒ𝐴𝑖1,….(2.12)

Generally,
ℒ𝑢𝑖(𝑗+1)=1(𝑡)𝑠ℒ𝑅𝑢1𝑗,…,𝑢𝑛𝑗+1𝑠ℒ𝐴𝑖𝑗.(2.13)

Applying the inverse Laplace transform to (2.10) gives the initial approximation𝑢𝑖0(𝑡)=ℒ−1ℋ𝑖(𝑠),𝑖=1,…,𝑛.(2.14)
Substituting these values of 𝑢𝑖0 into the inverse Laplace transform of (2.11) gives 𝑢𝑖1. The other terms 𝑢𝑖2,𝑢𝑖3,… can be obtained recursively in similar fashion from𝑢𝑖(𝑗+1)(𝑡)=ℒ−11𝑠ℒ𝑅𝑢1𝑗,…,𝑢𝑛𝑗+1𝑠ℒ𝐴𝑖𝑗,𝑗=0,1,2,….(2.15)

To provide clearly a view of the analysis presented above, three illustrative systems of pantograph equations have been used to show the efficiency of this method.

3. Test Problems

All iterates are calculated by using Matlab 7. The absolute errors in Tables 1–3 are the values of |𝑢𝑖∑(𝑡)−𝑛𝑗=0𝑢𝑖𝑗(𝑡)|, those at selected points.

Example 3.1. Consider the two-dimensional pantograph equations:
𝑢1=𝑢1(𝑡)−𝑢2(𝑡)+𝑢1𝑡2−𝑒𝑡/2+𝑒−𝑡,𝑢2′=−𝑢1(𝑡)−𝑢2(𝑡)−𝑢2𝑡2+𝑒𝑡/2+𝑒𝑡𝑢1(0)=1,𝑢2(0)=1.(3.1)
Applying the result of (2.14) gives us
𝑢10(𝑡)=4−2𝑒𝑡/2−𝑒−𝑡,𝑢20(𝑡)=2−2𝑒−𝑡/2+𝑒𝑡.(3.2)
The iteration formula (2.15) for this example is
𝑢1(𝑗+1)=ℒ−11𝑠ℒ𝑢1𝑗(𝑡)−𝑢2𝑗(𝑡)+𝑢1𝑗𝑡2,𝑢2(𝑗+1)=ℒ−11𝑠ℒ−𝑢1𝑗(𝑡)−𝑢2𝑗(𝑡)−𝑢2𝑗𝑡2.(3.3)
Starting with an initial approximations 𝑢10(𝑡) and 𝑢20(𝑡) and use the iteration formula (3.3). We can obtain directly the other components as
𝑢11(𝑡)=14+6𝑡+𝑒−𝑡−𝑒𝑡−2𝑒−𝑡/2−4𝑒𝑡/2−8𝑒𝑡/4,𝑢21(𝑡)=12−8𝑡−𝑒−𝑡−𝑒𝑡−4𝑒−𝑡/2+2𝑒𝑡/2−8𝑒−𝑡/4,𝑢12(𝑡)=158+16𝑡+172𝑡2−2𝑒−𝑡−14𝑒𝑡/2−6𝑒−𝑡/2−48𝑒𝑡/4−24𝑒−𝑡/4−64𝑒𝑡/8,u22(𝑡)=158+16𝑡+172𝑡2−2𝑒−𝑡−14𝑒𝑡/2−6𝑒−𝑡/2−48𝑒𝑡/4−24𝑒−𝑡/4−64𝑒𝑡/8.(3.4)Table 1 shows the absolute error of LDA with 𝑛 = 2, 4, and 6.

Example 3.2. Consider the system of multipantograph equations:
𝑢′1(𝑡)=−𝑢1(𝑡)−𝑒−𝑡𝑡cos2𝑢2𝑡2−2𝑒−(3/4)𝑡𝑡cos2𝑡sin4𝑢1𝑡4,𝑢2(𝑡)=𝑒𝑡𝑢21𝑡2−𝑢22𝑡2,𝑢1(0)=1,𝑢2(0)=0.(3.5)
Let us start with an initial approximation:
𝑢10𝑢(𝑡)=1,20(𝑡)=0.(3.6)
The iteration formula (2.15) for this example is
𝑢1(𝑗+1)=ℒ−11𝑠ℒ𝑢1𝑗(𝑡)−𝑢2𝑗(𝑡)+𝑢1𝑗𝑡2,𝑢2(𝑗+1)=ℒ−11𝑠ℒ𝑒𝑡𝐴1𝑗−𝐴2𝑗,(3.7)
where
𝐴𝑖0=𝑢2𝑖0𝑡2,𝐴𝑖1=2𝑢𝑖0𝑡2𝑢i1𝑡2,𝐴𝑖2=𝑢2𝑖1𝑡2+2𝑢𝑖0𝑡2𝑢𝑖2𝑡2,𝐴𝑖3=2𝑢𝑖1𝑡2𝑢𝑖2𝑡2+2𝑢𝑖0𝑡2𝑢𝑖3𝑡2,…,𝑖=1,2.(3.8)Table 2 shows the absolute error of LDA with 𝑛=1,2, and 3.

The iteration formula (2.15) for this example is𝑢1(𝑗+1)=ℒ−11𝑠ℒ2𝑢2𝑗𝑡2+𝑢3𝑗,𝑢(𝑡)2(𝑗+1)=ℒ−11𝑠ℒ−2𝐴2𝑗,𝑢3(𝑗+1)=ℒ−11𝑠ℒ𝑢2𝑗(𝑡)−𝑢1𝑗,(𝑡)(3.11)
where𝐴20=𝑢220𝑡2,𝐴21=2𝑢20𝑡2𝑢21𝑡2,𝐴22=𝑢221𝑡2+2𝑢20𝑡2𝑢22𝑡2,𝐴23=2𝑢21𝑡2𝑢22𝑡2+2𝑢20𝑡2𝑢23𝑡2,⋮(3.12)

4. Conclusion

The main objective of this paper is to adapt Laplace decomposition algorithm to investigate systems of pantograph equations. We also aim to show the power of the LAD method by reducing the numerical calculation without need to any perturbations, discretization, or/and other restrictive assumptions which may change the structure of the problem being solved. LDA method gives rapidly convergent successive approximations through the use of recurrence relations. We believe that the efficiency of the LDA gives it a much wider applicability.

References

J. R. Ockendon and A. B. Tayler, “The dynamics of a current collection system for an electric locomotive,” Proceedings of the Royal Society of London A, vol. 322, no. 1551, pp. 447–468, 1971.View at Google Scholar